# Covariance on historical data

An analyst gathered the following data for Stock A and Stock B: Time Period Stock A Returns Stock B Returns 1 10% 15% 2 6% 9% What is the covariance for this portfolio? A) 2. B) 3. C) 6. D) 12. Your answer: D was incorrect. The correct answer was C) 6. The formula for the covariance for historical data is: cov1,2 = {Ó[(Rstock A - Mean RA)(Rstock B - Mean RB)]}/n Mean RA = (10 + 6)/2 = 8, Mean RB = (15 + 9)/2 = 12 Here, cov1,2 =[(10-8)(15-12) + (6-8)(9-12)]/2 = 6 --------------------------------------------------------------------------------------------------------- An analyst observes the following return behavior between stocks X and Y. Time Period X’s Return Y’s return 1 7 5 2 9 8 3 10 11 4 10 8 What is the covariance of returns between stocks X and Y? A) + 2.75. B) - 1.50. C) + 2.25. D) + 1.50. Your answer: C was correct! Covariance = (1/n) * [Summation over t=1 of (ReturnX– MeanX) * (ReturnY – MeanY)] MeanX = (7+9+10+10)/4 = 9; MeanY = (5+8+11+8)/4 = 8 CovX,Y = [(7-9)(5-8)+(9-9)(8-8)+(10-9)(11-8)+(10-9)(8-8)] / 4 = 2.25 --------------------------------------------------------------------------------------------------------- Looking at these questions from schweser is just frustrating, they elected to divde by n instead of n-1, isn’t this gonna introduce bias into the calculation? The way I understand it is you divide by n-1 to allow the estimate to be unbiased, did schweser screw up again or am I missing something here?

liaaba - The rule is to use n when analyzing a population. Use n-1 when analyzing a sample. However, I struggle with the decision to classify the data as a sample or a population. I asked a similar question a couple days ago… here is the thread of responses. http://www.analystforum.com/phorums/read.php?11,613338,613535#msg-613535 I am still unclear on why this data represents a population as opposed to a sample. I guess it is safe to assume the means are known… therefore, we divide by n… therefore a population.

“An analyst observes the following return behavior between stocks X and Y.” “An analyst gathered the following data for Stock A and Stock B” this can’t be a population, to be a population, you have to observe EVERYTHING there is, so since the existence of company X and Y (or A and B), which the question did not specify, unless you make the assumption that whatever is included in the question represents their entire history unless this is one of those questions written by those omniscient question writters as Joey refers to them as.

It’s not a population issye; it’s wether the means are known. You had to calculate them so I guess they aren’t known

ok, i’m not for taking shortcuts usually, but here’s the BA2 plus shortcut since you L1’ers are getting close to crunch time. Hit 2nd DATA (the 7 key) In x1 enter 10, hit the enter key (like you do on your cash flow stuff) hit the down arrow. in y1 type in 15, enter… etc. Once you plug in all of the numbers, hit 2nd STAT (over the 8 key) start looking at the info it gives you- mean, variance, std dev, and also r. think to your formula r = cov (x,y)/ std dev X x std dev Y. you have everything here except cov 6 = cov/ 2 x 3 magic- saves you the manual mess ups on the test if ever you get one of these and takes you about 30 seconds. HOWEVER- make sure you understand the stuff before you start taking shortcuts because when you on the test get some definition questions and all you know is plug and chug on your calulator, you’re dead. Hope this helps- the math part is really easy w/ the aid of your CFA approved calculator. I have no idea how to do it on HP- i can’t even add 2 numbers, but i’m sure someone can walk you through it if you have that calculator. explore your calculator functions- they can save you a lot of time. good luck all.

ya, I know it’s a question of whether the population mean is known, cos you’re using it in the calculation of the variance estimator, but I think what apcarlso was thinking of is probably if you have the population, then you can use the all the members given in the population to calculate the population mean, but unless the question specifically states whether it’s a population or sample, it’s basically a crapshoot… I just hope on the actual exam they would be more specific about whether they’re providing a population or the sample… bannisja, ya, I use the stats worksheet too, but the problem is, they give you both the sample std dev and the population std dev, so it still boils down to whether you know to use the sampel ones or the population ones…

It’s just not true - in particular, the population of stock returns is something like [-100%, Infinity) and you have no idea what the distribution is on that (probably).

I’m not saying what they given represents a population, in fact, I used them as sample when I did those questions, but just that the solution from schweser treated them as sample, so even if we treated them correctly as sample, you can still get them wrong, for schweser anyway, and I was just saying I hope the actual exam is not like this

The actual exam is probably not like that, but I am less and less sure taht anybody at CFAI knows anything about this stuff. I send them errata weekly and I don’t even have the books. The send nice mails back saying oops we made another mistake…

haha, to be honest, this is the second time I’m going through the CFAI text, I’m noticing more and more typos in various sections of the readings… just like that GIPS compliance requirement thing I posted a few days ago, “up to … a minimum”, you figure they have someone proof read it…

-------------------------------------------- An analyst observes the following return behavior between stocks X and Y. Time Period X’s Return Y’s return 1 7 5 2 9 8 3 10 11 4 10 8 What is the covariance of returns between stocks X and Y? A) + 2.75. B) - 1.50. C) + 2.25. D) + 1.50. ----------------------------------------------------------------------------- If you do this as Cov(x,y) = Corr(x,y) * std(x) * std(y), and use your calculator to compute Corr(x,y), the answer you shoud get is 3. The calculator assumes that you use n-1, instead of n. There is no way to tell unless they tell you whether it is population or sample. Dreary

That’s just wrong. 1) That’s not the issue with populations and samples. 2) The population for stock returns is also possible increments of some stochastic process, in most probability books a geometric Brownian motion process with the caveat that it’s really more complicated than that. To identify a population value they would say the “mean return over each time increment is [blah]” because they have God-like omniscience. You need to read again what a population is and what a smaple is - they are wildly different beasts and you should always be able to tell one from the other.

That’s not helpful at all. What people are asking here is whether to use “n” or “n-1”. You are basically suggesting that on the exam, they should state that “…the mean return over each time increment is [blah]”, or that we should be aware of a stochastic process, and a geometric Brownian motion process? Holy cow! Come on, the short answer is that we should all hope that the CFAI questions will tell us whether it is a population or a sample data, or they would state it in a way that says so. None of the LOS’s says that you will be quizzed on advanced population versus sample identification techniques… Dreary

I agree with Dreary. They will make it clear on the exam for sure. Also, when you refer back to the initial question, it refers to the portfolio’s first and second periods. This implies to me that these are the only (first and second) two periods for the portfolio (i.e. population = n not (n-1)). You could argue this till the cows come home. I don’t think there is a clear cut answer. Just know the difference between the two, read the questions carefully - that’s all you can do irrespective of whether they state the differences or not.

Hunter007 Wrote: ------------------------------------------------------- > I agree with Dreary. They will make it clear on > the exam for sure. > > Also, when you refer back to the initial question, > it refers to the portfolio’s first and second > periods. This implies to me that these are the > only (first and second) two periods for the > portfolio (i.e. population = n not (n-1)). > > You could argue this till the cows come home. I > don’t think there is a clear cut answer. Just know > the difference between the two, read the questions > carefully - that’s all you can do irrespective of > whether they state the differences or not. --Hunter007, honestly, I think schweser just messed up the answers to those 2 questions, there’s no way you could’ve been able to tell those 2 were populations, I think Joey is right about if they want to tell you it’s a population, they should give you the population mean/variance, whichever it is you need to do the math, but when you have to calculate it yourself, it’s highly unlikely it’s a population, unless they explicitly say so… these 2 questions come from the port man section from schweser, I don’t recall running into messed up problems like this when I was doing quants, so I’m hoping schweser just messed up these 2 questions…

Dreary and Hunter - You’re just wrong. They will tell you what the mean is if you should use n and they will tell you what X-bar is if you use n-1. There really is no ambiguity unless you work very hard to create it. The absolutely will not say "use sample covariance’ or ‘use population covariance’ or some weird thing like that. You have to learn to tell the difference.

@ Dreary when u plug all the numbers form BA calc, the answer u get is +2.24982999 which is pretty close to +2.25, the correct answer. If you are really confused try using both sample and population st.dev and see which answers fits better, that’s the beauty of mc questions.

FisherSU, I assume you used the stats worksheet, and since it doesn’t give you cov by default, I assume you used r (correlation coefficient), std dev, but here’s the problem, BA calc gives you both the sample std dev and the population std dev, so, depending on which set you used, you could get both 2.25 and 3…

FisherU, on the HP the answer is 2.99999. Dreary

And the answer really is 3 and Schweser messed up another statistics problem. They mess up all their statistics problems. There is nobody at Schweser who has a clue about stats and probs.